# Tagged Questions

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### Shortest distance between two curves

Let $C_1= \{ (x, y) \in \mathrm{R}^2 : y = x^2 +1 \}$ and $C_2= \{ (x, y) \in \mathrm{R}^2 : x = y^2 +1 \}$, find the points which minimize distance between $C_1$ and $C_2$. What I tried is: we know ...
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### Multivariable optimization books

I have some economic data in hand, and I would like to make forecasting out of it (e.g., consumer demand, price elasticity and so on). As far as I understand, these characteristics can be (to some ...
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### Lagrangian Multiplier Question

I can do question 2 easily but I'm running into some problems proving 1 rigorously. No idea how to go about doing it at all.
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### Optimization with a few Variables (AMC 12 question)

In the 2013 AMC 12B, question 17 says: Let $a$,$b$, and $c$ be real numbers such that $a+b+c=2$, and $a^2+b^2+c^2=12$ What is the difference between the maximum and minimum possible values of $c$? ...
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### Extermum under constraint of parabula

Find the closet point on $2x^2-4xy+2y^2-x-y=0$ to the line $9x-7y+16=0$. Hint: the distance between $(x_0,y_0)$ to $ax+by+c=0$ is $d = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$. For using lagtrange ...
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### Prove that a multivariable function doesn't have global extremes

So my question is actually this. Say I have a function $F:\mathbb R^2\to\mathbb R$. If I find all the potential local extremes by finding the roots of the partial derivatives and I find that only one ...
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### Using Lagrange multipliers to maximize function

Use Lagrange multipliers to maximize function $$f(x,y)=6xy,$$ subject to the constraint $$2x+3y=24.$$ $$F(x,y,\lambda)=6xy+\lambda(2x+3y-24)$$ $$F_{x}=6y+2\lambda=0$$ $$F_{y}=6x+3\lambda=0$$ ...
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### Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2*(3ab^3+12a^3b-6a^3b^2)*\sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of $a$ and $b$ that ...
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### Finding extrema of multivariable functions.

A problem asks me to find the absolute extrema of the function given by $f: \mathbb{R}^2 \rightarrow \mathbb{R} ,f(x,y)=(x^2+y^2)e^{-(x^2+y^2)}$. Now, how can I find the critical points?. As far as I ...
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### Is the hessian negative semi-definite if we have an interior maximum?

Is it true that given a smooth scalar field f on a domain D , if f attains a maximum (minimum) on the interior of D then the hessian of f evaluated at this max (min) is negative (positive) ...
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### Constrained optimisation question

Since $f$ has a local extremum at $x_1$, then surely the LHS of equation (3) always zero? If so, then isn't lambda always simply zero too? But this cannot be, otherwise the last sentence of the ...
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### Maximizing a sum of inner products

Someone asked this question on a French maths forum here and it caught my attention. The question is the following: let $(E, \langle \cdot, \cdot \rangle)$ be a Euclidean vector space. Find the ...
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### How do I find out if a critical point of a function is a maximum or a minimum?

If I've found the critical point of a function defined in some constraint (perhaps using Lagrange multipliers and the like); how do I find out if it's a relative/global maximum/minimum of a function ...
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### Jacobian in Levenberg-Marquardt for 4-Parameter equation

I am trying to fully understand how I can use Levenberg-Marquardt to minimise a 4 parameter equation. There are lots of fancy programs to do this but the documentation about the mathematics is ...
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### Does the Lagrange multiplier method always give a saddle point problem?

I am considering the following problem: $$\min_u J(u)\text{, s.t. } \, H(u)=0.$$ Use Lagrange multiplier method, then $L(u, \lambda)=J + \lambda H(u)$. Does the critical point of the $L$ always ...
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### Optimising function of two variables with Lagrange multipliers

I am trying to find formulae for the $x$ and $y$ that maximise the function $f(x,y)=a(x + p)^bc(y + q)^d$, subject to the constraints: $$x \geq 0$$ $$y \geq 0$$ $$x + y + p + q \leq M$$ Where $a$, ...
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### Deducing that a matrix is indefinite using only its leading principal minors

$A$ is indefinite iff $A$ fits none of the above criteria. Equivalently, $A$ has both positive and negative eigenvalues. Also equivalently, $x^TAx$ is positive for at least one $x$ and ...
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### The 2nd total derivative (Hessian) of a composite function -Version 1

Let $f\in C^2(\mathbb R^n,\mathbb R)$ and $Df:A\subset\mathbb R^n\to L(\mathbb R^n,\mathbb R)$ so that $Df_x:\mathbb R^n\to\mathbb R$ is $f$'s total derivative at $x\in\mathbb R^n$. ...
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### Why does determining the nature of local extrema for $\mathbb R \to\mathbb R$ functions require twice continuous-differentiability?

In the text Elementary Classical Analysis, why does Marsden specify the condition "twice continuously differentiable" here? Isn't mere twice-differentiability sufficient for the purpose indicated? ...
How do I compute the global minimum or maximum of the function $f(x,y)=-\sin x\cos y$. Given it is on a square $(0\leq x\leq 2\pi)$ and $(0\leq y\leq2\pi)$